Fields

Complex Numbers

Definition: Complex numbers are numbers of the form a+bia+bi where a,bRa,b \in \mathbb{R} and ii is the imaginary unit such that i2=1i^2=-1. The set of complex numbers is denoted by C\mathbb{C}.

C={a+bia,bR}\mathbb{C} = \{a+bi \mid a,b \in \mathbb{R}\}

addition and multiplication are defined as follows:

(a+bi)+(c+di)=(a+c)+(b+d)i(a+bi)(c+di)=(acbd)+(ad+bc)i\begin{align*} (a+bi)+(c+di) = (a+c)+(b+d)i \\ (a+bi)(c+di) = (ac-bd)+(ad+bc)i \end{align*}

The set of real numbers is a subset of the complex numbers, i.e. RC\mathbb{R} \subset \mathbb{C}. The set of imaginary numbers is also a subset of the complex numbers, i.e. IC\mathbb{I} \subset \mathbb{C}.

Euler's formula is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that for any real number xx:

eix=cos(x)+isin(x)e^{ix} = \cos(x) + i \sin(x)

where ee is Euler's number, the base of natural logarithms, and ii is the imaginary unit, which satisfies the equation i2=1i^2 = −1.

The formula is still valid if xx is a complex number. In particular, if x=πx = \pi, Euler's formula states that:

eiπ+1=0e^{i\pi} + 1 = 0

The complex conjugate of a complex number z=a+biz = a + bi is given by zˉ=abi\bar{z} = a - bi. The complex conjugate of a complex number zz is denoted by zˉ\bar{z}.


#EE501 - Linear Systems Theory at METU